In a rectangle the length is 2 cm longer than the width. If its length and width are both increase by 4 cm it's area is increased by 56 cm. Find the original length and width.

Answer:

The correct option is 1. The image of (2,4) is (4,-8).

Step-by-step explanation:

The given rule is

[tex]R_x{\circ}D_{o,2}:(2,4)[/tex]

The transformations perform from right to left. [tex]D_{o,2}[/tex] means dilation with scale factor 2 and center of dilation is origin.

The given rule defines the dilation with scale factor 2 and center of dilation is origin followed by reflection across x-axis.

If a figure dilated by scale factor k and the center of dilation is origin, then

[tex](x,y)\rightarrow (kx,ky)[/tex]

The scale factor is 2,

[tex](x,y)\rightarrow (2x,2y)[/tex]

[tex](2,4)\rightarrow (4,8)[/tex]

If a figure reflected across x-axis, then x-coordinate remains the same but the sign of y-coordinate is changed.

[tex](x,y)\rightarrow (x,-y)[/tex]

[tex](4,8)\rightarrow (4,-8)[/tex]

Therefore image of (2,4) is (4,-8) and option 1 is correct.

Answer:

Volume of tray =[tex]4x^{3} -60x^{2} +216x[/tex]

Step-by-step explanation:

The Dimensions of rectangular cardboard is given by 12 inches by 18 inches

Length of the rectangular cardboard = 18 inches

Width of the rectangular cardboard = 12 inches

if the square of sides x inches is cut from each corner of the rectangular cardboard and folded to make a tray, then we have

Length of the tray = length of cardboard - 2 (side of the square )

                               = [tex]18-2x[/tex]

Width of the tray = width of cardboard - 2( side of square)

                            = [tex]12-2x[/tex]

height of tray = sides of square

                       = x

volume of tray = Length × width × height

Volume of tray = [tex]x(18-2x)(12-2x)[/tex]

first we multiply (12-2x) and (18-2x)

Volume of tray =[tex]x(18(12-2x)-2x(12-2x))\\[/tex]

                         =[tex]x(216-36x-24x+4x^{2})[/tex]

                         =[tex]x(4x^{2}-60x+ 216)[/tex]

                         =[tex]4x^{3} -60x^{2} +216x[/tex]

Hence the volume of tray =[tex]4x^{3} -60x^{2} +216x[/tex]

The volume of the Tray is given by the equation [tex]4x^3-60x^2+216x[/tex] and this can be determined by using the given data.

Given :

If the square is cut from each corner of the cardboard and folded to make a tray, then according to the given data:

Tray length = 18 - 2x

Tray width = 12 - 2x

Tray height = x

Now, the volume of the tray is given by the expression:

The volume of Tray = [tex]x(12-2x)(18-2x)[/tex]

Simplify, the above expression.

[tex]\rm Volume = (12x - 2x^2)(18-2x)[/tex]

[tex]\rm Volume = 216x -24x^2-36x^2+4x^3[/tex]

[tex]\rm Volume = 4x^3-60x^2+216x[/tex]

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Answer: C < B < A

Step-by-step explanation:

First, let's evaluate the growth rate of A from [1, 3]

[tex]f(x) = 200\bigg(\dfrac{3}{2}\bigg)^1[/tex]

    = 300

coordinate is (1, 300)

[tex]f(x) = 200\bigg(\dfrac{3}{2}\bigg)^3[/tex]

    = 675

coordinate is (3, 675)

Average growth rate (AGR) is: [tex]\dfrac{675-300}{3-1} = \dfrac{375}{2} = 187.5[/tex]

AGR (A) = 187.5

*************************************************************************************

Next, let's evaluate the growth rate of B from [1, 3]

The coordinates are already provided as (1, 120) and (3, 480)

Average growth rate (AGR) is: [tex]\dfrac{480-120}{3-1} = \dfrac{360}{2} = 180[/tex]

AGR (B) = 180

*************************************************************************************

Lastly, let's evaluate the growth rate of C from [1, 3]

f(x) = 600(1.2)ˣ

f(x) = 600(1.2)¹

     = 720

coordinate is (1, 720)

f(x) = 600(1.2)³

     = 1036.8

coordinate is (3, 1036.8)

Average growth rate (AGR) is: [tex]\dfrac{1036.8-720}{3-1} = \dfrac{316.8}{2} = 158.4[/tex]

AGR (C) = 158.4

Answer:

All explanations are correct

Step-by-step explanation:

In triangle ABC, CD is the perpendicular bisector of AB, thus using ΔADC and ΔBDC,

AC=BC ( Since CD is perpendicular to AB, therefore C is equidistant from both A and B)

∠ADC=∠BDC=90°(CD perpendicular AB)

CD=CD( Reflexive property)

therefore, by SAS rule of congruency,

ΔADC≅ ΔBDC,

Also, In the same triangles, AC=BC ( Since CD is perpendicular to AB, therefore C is equidistant from both A and B)

CD=CD( Reflexive property)

AD=BD (D is the midpoint and divides AB into two equal halves)

Thus, by SSS rule of congruency,

ΔADC≅ ΔBDC,

Thus, all the three explanations are correct.

In the given case, we can conclude that All explanations are correct

In triangle ABC, let CD be the perpendicular bisector of AB. We can use the properties of triangles to demonstrate that ΔADC is congruent to ΔBDC.

AC=BC: This holds true because CD is perpendicular to AB, making point C equidistant from both A and B.

∠ADC=∠BDC=90°: Since CD is perpendicular to AB, both angles ADC and BDC are right angles.

CD=CD: This is a reflexive property, stating that any line segment is equal to itself.

By applying the SAS (Side-Angle-Side) rule of congruency using the above properties, we can conclude that ΔADC is congruent to ΔBDC.

Furthermore, we can establish that in these congruent triangles:

AC=BC (since C is equidistant from A and B due to CD being perpendicular to AB).

CD=CD (reflexive property).

AD=BD (as D is the midpoint of AB).

Hence, using the SSS (Side-Side-Side) rule of congruency, we can also conclude that ΔADC is congruent to ΔBDC.

Therefore, all three explanations correctly demonstrate the congruence of these triangles.

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Answer:

Step-by-step explanation: To find a y intercept you must find what y is when x is 0 so because it already tells you on the table the y intercept is 4. To find slope take 2 x intercepts and 2 y intercepts. For this well take the 0,4 and 5,1. The take the y from both pairs and minus them so it will look like 4 minus 1. Then draw a line under the equation under this line do the same thing but with the x's from both pairs so 0-5 then you will get as a result of both -3/5 as your slope.

Answer:

slope: -3/5

y-intercept: (0, 4)

slope-intercept form: y = -3/5x + 4

Step-by-step explanation:

To find the slope of this line, you would take two points from the table and substitute their coordinates into the slope formula.

Slope formula: [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

I'm going to use the points (0, 4) and (5, 1). You can really use any point from the table. Substitute these points into the formula to find the slope.

(0, 4), (5, 1) → [tex]\frac{1-4}{5-0} \rightarrow \frac{-3}{5}[/tex]

This means the slope of the line is -3/5.

The y-intercept will always have the value of x be 0 (so the point is solely on the y-axis), so by looking at the table we can see that the y-intercept is at (0, 4).

Since we have the slope and a point of the line, we must use point-slope form to find the equation of the line in slope-intercept form. Substitute in the point (0, 4) --you could use any point from the table-- and the slope -3/5 into the point-slope form equation.

point-slope form: y - y1 = m(x - x1) --you'll be substituting the point coordinates and slope into y1, x1, and m.

y - (4) = -3/5(x - (0))

Simplify.

y - 4 = -3/5x

Add 4 to both sides.

y = -3/5x + 4 is the equation of the line in slope-intercept form (you have both the slope and the y-intercept in this form).

Answer:

[tex]8.05 * 10^{5}[/tex]

Step-by-step explanation:

Thinking process:

the population of Park Crest Middle School = 400 children

The population in the state = 806, 240 children

The difference:

[tex]806240 - 400\\= 805840[/tex]

Expressing the difference in standard form gives:

8.05 × 10⁵

Answer: An x and y value, parenthesis around the values, and a comma in between them.

Examples of ordered pairs: (2, 5), (1872, 1683), (-3, 7), (7/8, -10/2)

Answer:

4(3+x)

Step-by-step explanation:

If any thing is in the brackets you need to do it first

so if you do 3 + x × 4 it will give yo 4x + 3 or 3 +4x

The given expression 4 (x + 3) is equivalent to the expression 4x + 12.

The equivalent operations are those that have various forms but have the same outcome.

The expression is given below.

→ 4 (x + 3)

The expression can be written as

→ 4x + 12

The given expression 4 (x + 3) is equivalent to the expression 4x + 12.

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Answer:

70, 105, 140

Step-by-step explanation:

Answer:

x= 0.4

Step-by-step explanation:

We need to solve and find the value of x

7 1/3x = 1.6 ÷ 6/11

Converting mixed fraction into improper fraction

22/3 x = 1.6 ÷ 6/11

Converting ÷ sign by X and reciprocating the term

22/3 x = 1.6 * 11/6

Now divide both sides by 3/22

x = 1.6 * 11/6 * 3/22

x = 1.6 * 1/2 * 1/2

x = 1.6/4

x = 0.4

S0, x= 0.4

Answer:

[tex]y=cos(x+\pi )[/tex]

Step-by-step explanation:

From the graph, we can see that y = -1 when x  = 0.

So to check whether which of the given options is the equation of the given graph, we will set our calculator to the radian mode and then plug the value of x as 0.

1. y = cos(x + pi/2) = cos(0 + pi/2) = 0

2. y = cos(x+2pi) = cos(0+2pi) = 1

3. y = cos(x+pi/3) = cos(0+pi/3) = 1/2 = 0.5

4. y = cos(x+pi) = cos(0+pi) = -1

Therefore, the equation of this graph is y = cos(x+pi) = cos(0+pi) = -1.

Answer:

2/5 * pi radians = 72 degrees

Step-by-step explanation:

To convert radians to degrees, we multiply by 180/pi

2 * pi/5 * 180/pi =

The pi/pi cancels leaving

2/5 * 180

360/5

72

2/5 * pi radians = 72 degrees

Answer:

8 daisies

Step-by-step explanation:

You have to multiply the amount of vases by daisies.

Answer:

Givens

We know that Morita arranges 4 cases, each of them has 10 flowers.

So, if there are 2/10 daisies per case, that means each can contains 2 daisies.

But, she did 4 arrangements. There are 2x4 = 8 daisies in total.

Now, last weekend, Morita used 10 daisies to make flower arrangements, and we know she tend to use 2/10 of the flowers as daisies, that means this fraction represents daisies,

[tex]\frac{2}{10}x=10\\ x=\frac{100}{2}\\ x=50[/tex]

Therefore, there are 50 flowers in total.

the cheap answer is, we divide 500 by (2+3+5+10) and then distribute accordingly.

[tex]\bf \cfrac{500}{2+3+5+10}\implies \cfrac{500}{20}\implies \cfrac{25}{1}\implies 25 \\\\\\ \stackrel{2\cdot 25}{2}~~:~~\stackrel{3\cdot 25}{3}~~:~~\stackrel{5\cdot 25}{5}~~:~~\stackrel{10\cdot 25}{10}\qquad \implies \qquad 50~~:~~75~~:~~125~~:~~250[/tex]

Answer:

15%

Step-by-step explanation:

Percent means out of 100.  Lets get the denominator out of 100

3/20 * 5/5  = 15/100

This is 15%

Final answer:

To convert the fraction 3/20 to a percentage, first convert it to a decimal by dividing 3 by 20. Then, multiply the decimal by 100 to get the percentage.

Explanation:

To express 3/20 as a percentage, you can multiply the fraction by 100. The calculation is (3/20) * 100 = 15%. Therefore, 3/20 as a percentage is 15%. This means that 3 out of every 20 parts represent 15% of the whole. In percentage terms, it provides a convenient way to compare the fraction to 100, making it easier to understand its relative size or portion in relation to the total. In this case, 15% indicates the proportion of the whole that is represented by the fraction 3/20.

Answer:

g⁻¹(x) = ±√(x + 25)

Step-by-step explanation:

g(x) = x² - 25     Rename g(x) as y

  y = x² - 25

=====

Solve for x  

        y = x² - 25          Add 25 to each side

y + 25 = x²                  Take the square root of each side

       x = ±√(y + 25)     Switch x and y

       y = ±√(x + 25)     Rename y as g inverse

g⁻¹(x) = ±√(x + 25)

=====

See the graphs of g(x) and g⁻¹(x) below.

g(x) is the red line. g⁻¹(x) is the purple line.

Each graph reflects the other about the dashed line representing the function y = x.

Answer:

180 degrees in radians is π

Step-by-step explanation:

0° = 0 radians

90°= π/2

180 °= π

270°= 3π/2

360° = 2π

and so on

Answer:

B is the right one i bet u 100 point s

Answer:

B is the correct answer.

The brackets ║ represent absolute value. Absolute value is ALWAYS positive. I hope this helps you!

-Mikayla

Answer:

88 ft

Step-by-step explanation:  Start with the formula for the area of a rectangle:

A = L*W, where L is the length and W is the width.

Here, the width is W = 24 ft, and the area i s 480 ft^2.  Thus,

A = 480 ft^2 = L (24ft), which yields L = (480/24) ft = 20 ft.

The length of the fencing required is actually the perimeter of the yard, which is P = 2L + 2W.

Substituting 20 ft for L and 24 ft for W, we get:

P = perimeter of yard = amount of fending needed:

P = 2(20 ft) + 2(24 ft) = 88 ft

Ken would need 88 ft of fencing to enclose his entire yard.

Ken needs to buy 88 feet of fencing to enclose his rectangular yard, which has an area of 480 square feet and is 24 feet wide.

To find out how much fencing Ken would need to buy to enclose his rectangular yard, we first need to calculate the dimensions of the yard. We know the area of the yard is 480 square feet and the width is 24 feet.

To find the length, we use the following equation for the area of a rectangle (Area = length  imes width):

480 square feet = length x 24 feet

By dividing both sides by 24 feet, we obtain:

length = 480 / 24

length = 20 feet

Now, to calculate the total amount of fencing needed, we add together the lengths of all four sides:

Total fencing = 2  imes (length + width)

Total fencing =  2  imes (20 feet + 24 feet)

Total fencing = 2  imes 44 feet

Total fencing = 88 feet

Therefore, Ken needs to buy 88 feet of fencing to enclose his yard completely on all four sides.

Answer:

This is a direct variation where the constant of variation is 5/9

Step-by-step explanation:

The equation for direct variation is y = kx

5x-9y =0

Lets try to get this equation in the above form

Add 9y to each side

5x-9y +9y = 9y

5x = 9y

Divide each side by 9

5/9 x = 9y/9

5/9x = y

This is in the form y = k x where k =5/9

Answer:

2. m = b³ (= 216)

3. logp(x) = -4

Step-by-step explanation:

2. The given equation can be written using the change of base formula as ...

... log(m)/log(b) + 9·log(b)/log(m) = 6

If we define x = log(m)/log(b), then this becomes ...

... x + 9/x = 6

Subtracting 6 and multiplying by x gives ...

... x² -6x +9 = 0

... (x -3)² = 0 . . . . . factored

... x = 3 . . . . . . . . . value of x that makes it true

Remembering that x = log(m)/log(b), this means

... 3 = log(m)/log(b)

... 3·log(b) = log(m) . . . . . multiply by the denominator; next, take the antilog

... m = b³ . . . . . . the expression you're looking for

___

3. Substituting the given expression for y, the equation becomes ...

... logp(x^2·(p^5)^3) = 7

... logp(x^2) + logp(p^15) = 7 . . . . . use the rule for log of a product

... 2logp(x) + 15 = 7 . . . . . . . . . . . . . use the definition of a logarithm

... 2logp(x) = -8 . . . . . . . . . . . . . . . . subtract 15

... logp(x) = -4 . . . . . . divide by 2

Answer:

For #3: [tex]\log_px=-4[/tex]

Step-by-step explanation:

I'm a little rusty on my logarithm rules for #2, but here's an explanation of #3.

In a sense, we can think of operations like subtraction and division as different ways of representing addition and multiplication. For instance, the same relationship described by the equation 2 + 3 = 5 is captured in the equation 5 - 3 = 2, and 5 × 2 = 10 can be restated as 10 ÷ 2 = 5 without any loss of meaning.

Logarithms do the same thing for exponents: the expression [tex]2^3=8[/tex] can be expressed in logarithms as [tex]\log_28=3[/tex]. Put another way, logarithms are a sort of way of pulling an exponent out onto its own side of the equals sign.

Our problem gives us two facts to start: that [tex]log_p(x^2y^3)=7[/tex] and [tex]p^5=y[/tex]. With that, we're expected to find the value of [tex]\log_px[/tex]. [tex]p^5=y[/tex] stands out as the odd-equation-out here; it's the only one not in terms of logarithms. We can fix that by rewriting it as the equivalent statement [tex]log_py=5[/tex]. Now, let's unpack that first logarithm.

For a refresher, let's talk about some of the rules logarithms follow and why they follow them:

Product Rule: [tex]\log_b(MN)=\log_bM+\log_bN[/tex]

The product rule turns multiplication in the argument (parentheses) of a logarithm into addition. For a proof of this, consider two numbers [tex]M=b^x[/tex] and [tex]N=b^y[/tex]. We could rewrite these two equations with logarithms as [tex]\log_bM=x[/tex] and [tex]\log_bN=y[/tex]. With those in mind, we could say the following:

And we have our proof.

Exponent Rule: [tex]\log_b(M^n)=n\log_bM[/tex]

Since exponents can be thought of as abbreviations for repeated multiplication, we can rewrite [tex]\log_b(M^n)[/tex] as [tex]\log_b(M\times M\cdots \times M)[/tex], where M is being multiplied by itself n times. From there, we can use the product rule to rewrite our logarithm as the sum [tex]\log_bM+\log_bM+\cdots+\log_bM[/tex], and since we have the term [tex]\log_bM[/tex] added n times, we can rewrite is as [tex]n\log_bM[/tex], proving the rule.

With those rules in hand, we're ready to solve the problem. Looking at the equation [tex]\log_p(x^2y^3)=7[/tex], we can use the product rule to split the logarithm into the sum [tex]\log_p(x^2)+\log_p(y^3)=7[/tex], and then use the product rule to turn the exponents in each logarithm's argument into coefficients, giving the equation [tex]2\log_px+3\log_py=7[/tex].

Remember how earlier we rewrote [tex]p^5=y[/tex] as [tex]log_py=5[/tex]? We can now use that fact to substitute 5 in for [tex]log_py[/tex], giving us

[tex]2\log_px+3(5)=7[/tex]

From here, we can simply solve the equation for [tex]\log_px[/tex]:

[tex]2\log_px+15=7\\2\log_px=-8\\\\\log_px=-4[/tex]

Hey man, your answer is (A) I took the test!!!

The first 4 terms in the sequence are :

Firstly , let us learn about types of sequence in mathematics.

Arithmetic Progression is a sequence of numbers in which each of adjacent numbers have a constant difference.

[tex]\boxed{T_n = a + (n-1)d}[/tex]

[tex]\boxed{S_n = \frac{1}{2}n ( 2a + (n-1)d )}[/tex]

Tn = n-th term of the sequence

Sn = sum of the first n numbers of the sequence

a = the initial term of the sequence

d = common difference between adjacent numbers

Geometric Progression is a sequence of numbers in which each of adjacent numbers have a constant ration.

[tex]\boxed{T_n = a ~ r^{n-1}}[/tex]

[tex]\boxed{S_n = \frac{a( 1 - r^n ) }{1 - r}}[/tex]

Tn = n-th term of the sequence

Sn = sum of the first n numbers of the sequence

a = the initial term of the sequence

r = common ratio between adjacent numbers

Let us now tackle the problem!

Given:

a = 4

r = 0.2

Solution:

[tex]T_n = a ~ r^{n-1}[/tex]

[tex]T_1 = 4 \times 0.2^{1-1} = 4 \times 1 = 4[/tex]

[tex]T_2 = 4 \times 0.2^{2-1} = 4 \times 0.2 = 0.8[/tex]

[tex]T_3 = 4 \times 0.2^{3-1} = 4 \times 0.04 = 0.16[/tex]

[tex]T_4 = 4 \times 0.2^{4-1} = 4 \times 0.008 = 0.032[/tex]

Therefore , the first 4 terms in the sequence are :

Grade: Middle School

Subject: Mathematics

Chapter: Arithmetic and Geometric Series

Keywords: Arithmetic , Geometric , Series , Sequence , Difference , Term

Answer:

29.03 units

Step-by-step explanation:

x^2 = 10^2 + 19^2 - 2*19*20 cos 122

=   100 + 361 - 720 * -0.52992

= 842.54

x = 29.03

In combinatorics, we use the combination formula to calculate the number of ways Mrs. Wong can choose 2 books out of 14. The result is 91 ways.

The subject of this question is combinatorics, a topic in mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as those specified in this question. In this case, Mrs. Wong has 14 special objects (books) and wants to choose 2 out of them.

We use the combination formula in this scenario. The combination formula is given by C(n, r) = n! / [(n-r)!*r!], where n represents the total number of objects, r is the number of objects to choose, and '!' denotes factorial.

Substituting n as 14 and r as 2 into the formula, we get: C(14,2) = 14! / [(14-2)!*2!] = 91.

Therefore, Mrs. Wong can choose 2 books out of 14 in 91 different ways.

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The question is about combinations in mathematics. Using the formula for combinations, we find that Mrs. Wong can choose 2 books out of 14 in 91 different ways.

This is a problem of combinations in mathematics. Mrs. Wong can choose two out of 14 unread books in a certain number of ways, and we're tasked to find that number.

Considering that order of selection does not matter, we can use the formula for combination: nCr = n! / r!(n-r)!. Here, 'n' is the total number of items, 'r' is the items to be chosen.

So the combinations she can make, denoted as 14C2, can be calculated like this:

14C2 = 14! / 2!(14-2)! = (14*13) / (2*1) = 91 ways

Therefore, Mrs Wong can choose 2 books out of 14 in 91 ways.

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Answer:

The altitude of the trapezoid is 60 feet.

Step-by-step explanation:

Trapezoid

Sum of the bases: B+b=280 feet

Area of the trapezoid: A=8,400 feet^2

Altitude of the trapezoid: h=?

The formula to calculate the area of a Trapezoid is:

A = (1/2) (B+b) h

Replacing the given values:

8,400 feet^2 = (1/2) (280 feet) h

(1/2) (280 feet) = (140 feet)

8,400 feet^2 = (140 feet) h

Solving for h: Dividing both sides of the equation by 140 feet:

8,400 feet^2 / (140 feet) = (140 feet) h / (140 feet)

60 feet = h

h = 60 feet

Answer: The altitude of the trapezoid is 60 feet.

Answer:

60

Step-by-step explanation:

Answer:

0.49 m²

Step-by-step explanation:

The area (A) is calculated using A = πr² ( r is the radius )

A = external area - internal area

external radius = [tex]\frac{1.2}{2}[/tex] = 0.6

internal radius = [tex]\frac{0.9}{2}[/tex] = 0.45

A = π(0.6² - 0.45²)

   = π(0.36 - 0.2025) = 0.1575π ≈ 0.49 m²

Final answer:

The area of the washers is 0.49m².

Explanation:

To find the area of the washers, we need to find the areas of the outer and inner circles and then subtract the smaller area from the larger one. The area of a circle is given by the formula A = πr², where r is the radius of the circle. In this case, the external diameter is 1.2m, so the radius is half of that, or 0.6m. The area of the outer circle is A₁ = π(0.6)² = 1.13m². Similarly, the radius of the inner circle is 0.45m, and the area of the inner circle is A₂ = π(0.45)² = 0.64m². Finally, the area of the washer is the difference between the two areas: A = A₁ - A₂ = 1.13m² - 0.64m² = 0.49m².